Up: Basic Coding Topology for
Choose an arbitrary codeword c from C. Counting the number of vectors within
S, we have c itself and all the vectors of distance 1 from c. Since c has positions, using our definition of metric, we have that there are precisely vectors of distance 1 from c. From Theorem 1.3, we know these spheres are mutually disjoint, therefore exactly
vectors of V are covered by the spheres of C. Thus for a 1-error correcting code, C is perfect if and only if
, since that's how many vectors are in V.
Any perfect 1-error correcting code C V has a length of the form , where is an integer.